Show that

We have,

$f(x)=x+\cos x-a$

$f^{\prime}(x)=1-\sin x=\frac{2 \cos ^{2} x}{2}$

Now,

$x \in R$

$\Rightarrow \frac{\cos ^{2} x}{2}>0$

$\Rightarrow \frac{2 \cos ^{2} x}{2}>0$

$\Rightarrow f^{\prime}(x)>0$

Hence, $f(x)$ is an increasing function for $x \in R$